Accelerometers used in guidance, navigation, and control systems, have to meet performance specifications in spite of structural and acoustic vibration environments. These systems typically output accelerometer values at relatively slow data rates, on the order of 100 Hz and slower. This is sufficient for aircraft navigation or missile guidance and control. Structural and acoustic vibrations, on the other hand, are typically much higher, in the 100 to 100,000 Hz range.
The average output from an accelerometer, taken over enough samples, would ideally be zero in a vibration environment as described above. The vibration environment being equally positive and negative in direction, with an average acceleration of zero, and no net change in velocity.
However, real accelerometers do not respond identically to positive and negative accelerations. That is, their output is not perfectly linear over the + and − range. As a result, their average output does not average to zero under vibration. Instead, they suffer a bias offset in vibration, an error that is referred to as vibration rectification error, or VRE. VRE is typically a significant problem for precision accelerometers in guidance, navigation, and control systems.
One source of accelerometer nonlinearity contributing to VRE is called cross coupling sensitivity. This refers to changes in the primary input axis sensitivity of the accelerometers as a function of cross axis accelerations. In particular, the cross coupling coefficient Kip (input axis sensitivity coupling with pendulous axis input) can be very large, and contributes significantly to nonlinearity and to VRE.
In pendulous vibrating beam accelerometers, the cross coupling coefficient Kip comes from two sources. First, the pendulum displaces under acceleration, causing the center of mass to move with respect to the supporting flexures or pivot. This causes a change in pendulous axis sensitivity, which then by definition is a cross coupling sensitivity Kip. This source for Kip is typically referred to as pendulum droop.
A second source for Kip is from the nonlinear force-frequency relationship in the vibrating beam force sensor. (The terms “vibrating beam force sensor”, “force sensor”, and “resonator” are used interchangeably). Because of this nonlinearity, input axis accelerations change pendulous axis sensitivity, and vice versa, resulting in Kip by definition.
FIG. 1 shows a prior art vibrating beam accelerometer with the orientation of vibrating beam force sensors (resonators 34-1, 34-2) relative to the pendulous proof mass 30 attached to a structure and stable member 38 via flexure(s) 32, such that:
1) Droop Kip is positive. That is, for positive accelerations along an input axis 40, the angular droop of the pendulum will increase the sensitivity along a pendulous axis 42.
2) Vibrating beam Kip is also positive. That is, for positive accelerations along the pendulous axis 42, both resonators 34-1, 34-2 go into compression, which by the nonlinear force-frequency relationship of the resonator, will increase the input axis sensitivity.
In summary, Kip nonlinearity results in accelerometer bias errors in vibration environments (VRE). Kip in vibrating beam pendulous axis accelerometers is driven both by pendulum droop and by the nonlinear force-frequency behavior of the vibrating beam force sensor.